This article presents an efficient domain decomposition algorithm of Schwarz waveform relaxation type for singularly perturbed Gierer-Meinhardt type nonlinear coupled systems of parabolic problems where the diffusion terms in each equation are multiplied by small parameters of different magnitudes. The magnitude of these small parameters leads to the sharpness and boundary layer behavior in the solution components. Our present algorithm considers a suitable decomposition of the domain and decouples the process of approximating the solution components at each time level. There are two different schemes proposed in this work. Specifically, the schemes use the backward Euler method combined with a suitable component-wise splitting for time discretization, while employing the central difference scheme for spatial discretization. The two numerical schemes differ in their splitting methods: Scheme 1 employs a Jacobi-type split, whereas Scheme 2 utilizes a Gauss-Seidel-type split. The exchange of information between neighboring subdomains is achieved through piecewise-linear interpolation. The convergence analysis of the algorithm is demonstrated using some auxiliary nonlinear systems. It is shown that the present procedure provides uniformly convergent numerical approximations to the solution having sharp spike components. Numerical experiments demonstrate that the considered algorithm with present discretization is more efficient in terms of accuracy and iteration counts than with the standard available approaches.

In this paper, we provide a theoretical analysis for a preconditioned steepest descent (PSD) iterative solver that improves the computational time of a finite difference numerical scheme for the Cahn-Hilliard equation with Flory-Huggins energy potential. In the numerical design, a convex splitting approach is applied to the chemical potential such that the logarithmic and the surface diffusion terms are treated implicitly while the expansive concave term is treated with an explicit update. The nonlinear and singular nature of the logarithmic energy potential makes the numerical implementation very challenging. However, the positivity-preserving property for the logarithmic arguments, unconditional energy stability, and optimal rate error estimates have been established in a recent work and it has been shown that successful solvers ensure a similar positivity-preserving property at each iteration stage. Therefore, in this work, we will show that the PSD solver ensures a positivity-preserving property at each iteration stage. The PSD solver consists of first computing a search direction (which requires solving a constant-coefficient Poisson-like equation) and then takes a one-parameter optimization step over the search direction in which the Newton iteration becomes very powerful. A theoretical analysis is applied to the PSD iteration solver and a geometric convergence rate is proved for the iteration. In particular, the strict separation property of the numerical solution, which indicates a uniform distance between the numerical solution and the singular limit values of $±1$ for the phase variable, plays an essential role in the iteration convergence analysis. A few numerical results are presented to demonstrate the robustness and efficiency of the PSD solver.

In this paper, we propose a linear, fully decoupled and unconditionally energy-stable discontinuous Galerkin (DG) method for solving the tumor growth model, which is derived from the variation of the free energy. The fully discrete scheme is constructed by the scalar auxiliary variable (SAV) for handling the nonlinear term and backward Euler method for the time discretization. We rigorously prove the unconditional energy stability and optimal error estimates of the scheme. Finally, several numerical experiments are performed to verify the energy stability and validity of the proposed scheme.

A model parabolic linear partial differential equation in a geometrical multi-scale domain is studied. The domain consists of a two-dimensional central node, and several one-dimensional outgoing branches. The physical coupling conditions between the node and the branches are either continuity of the solution or continuity of the normal flux. An iterative Schwarz method based on Robin transmission conditions is adjusted to the problem in each case and formulated in substructured form. The convergence of the method is stated. Numerical results when the method is used as preconditioner for a Krylov method (GMRES) are provided.

This paper studied the stability and superconvergence of a special isoparametric bilinear finite volume element scheme for anisotropic diffusion problems over quadrilateral meshes, where the scheme is obtained by employing the edge midpoint rule to approximate the line integrals in classical $Q_1$-finite volume element method. It can be checked that the scheme is identical to the standard five-point difference scheme for a special case. By element analysis approach, we suggest a sufficient condition to guarantee the stability of the scheme. This condition has an analytic expression, which covers the traditional $h^{1+\gamma}$-parallelogram and some trapezoidal meshes with any full anisotropic diffusion tensor. Moreover, based on the $h^2$-uniform quadrilateral mesh assumption, we proved the superconvergence $|u_I−u_h|_1=\mathcal{O}(h^2),$ where $u_I$ is the isoparametric bilinear interpolation of exact solution $u,$ and $u_h$ is the numerical solution. As a byproduct, we obtained the optimal $H^1$ and $L^2$ error estimates. Finally, the theoretical results are verified by some numerical experiments.

This paper concerns splitting methods for solving backward stochastic differential equations (BSDEs). By splitting the original $d$-dimensional BSDE into $d$ BSDEs and approximating these split BSDEs, we propose splitting schemes for the BSDE. The splitting schemes are rigorously analyzed and first-order error estimates are theoretically obtained. Numerical tests are given to verify the theoretical results.